Analysis of special aspects of the integrodifferential equation system describing biomedical objects' condition with ionic conduction

Background. The topicality of building and researching the features of the nonlinear integrodifferential equation systems is characterized by their ability to describe complex physicochemical (biomedical) dissipative structures, i.e. the so-called Turing's models. Biomedical objects with ionic conduction are an important special case of these models. The status monitoring of that objects for the purpose of obtaining new abilities for the medical trial and analysis is a special practical interest. Many ways of monitoring and researching are expressed by the inverse coefficient problem. The goal of this article is to theoretically analyze special aspects of the integrodifferential equation system, which have been introduced in the previous papers with respect to particular characteristics of status modeling of biomedical objects with ionic conduction, and to formulate the inverse problems on the basis thereof. Materials and methods. Theoretical and practical facts of the experimental methods of analytic chemistry based on the electrochemical analysis (chronopotentiometry, voltammetry, polarography etc.) were used to understand the special aspects of the nonlinear integrodifferential equation system. Specifics of the geometry of an electrochemical cell (electrodes, volumes of systems, special structures) and measure procedure modes, as well as electrochemical reactions were taken into account. The asymptotic approach in the form of a stationary case of researching the equation system with plane condenser geometry was examined in the studies of problems of dissipative structures modeling. The specifics of the inverse problems formulation by means of the integrodifferential equation system were determined in terms of the standard incorrect problem theory. Results. A link between the aspects of specific electrochemical cell geometry and a key integral item from the nonlinear integrodifferential equation system were analyzed. The contribution of researching the equation system into unification of different electrochemical and physical phenomena, separated by significant localization thereof in electrochemical cell geometry, was established. The authors determined the drawbacks of modeling by the equation system for continuous environment: it is impossible to naturally conduct the processes of sorption and desorption on electrode surface. The authors suggested a method which makes it possible to partially evade the drawback and also to describe the electrode volume charge texture and its mutual effect with electrolyte ions. The equation system for a simple two-component reaction $O+z e^- \frac{\overrightarrow{k_e}} {\overleftarrow{k_e}} R$ with boundary conditions and nonlinear members was comprised including a stationary case. Conclusions. One of the most important terms of the equation system is the integral members based on the Green's function, even the simplified form of which could describe the double-electric layer near the electrode surface. Its more precise and strict building, in perspective, links the integrodifferential equation system with the phenomenon of sorption-desorption and interaction of the electrode bulk with ions and molecules into electrolyte. The built system, based on simple cases of chemical reaction and electrochemical cell geometry, showed that the stationary limit is a case of dissipation structures which can't be obtained by means of elementary linear equations from the classical course of electrochemistry. The invers problem for the system under investigation is formulated in a standard form except that there is a special aspect in the shape of availability of an integral in the Fredholm's form, not with singular and weakly singular as it complies with the standard form of the Green's kernel.